Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Bounds on the Largest of Minimum Degree Laplician Eigenvalues of a Graph

Abstract: In this paper we give three upper bounds for the largest of minimum degree Laplacian eigenvalues of a graph and also obtain a lower bound for the same.

Spectral graph theory of brain oscillations -- revisited and improved

Abstract: Mathematical modeling of the relationship between the functional activity and the structural wiring of the brain has largely been undertaken using non-linear and biophysically detailed mathematical models with regionally varying parameters. While this approach provides us a rich repertoire of multistable dynamics that can be displayed by the brain, it is computationally demanding. Moreover, although neuronal dynamics at the microscopic level are nonlinear and chaotic, it is unclear if such detailed nonlinear models are required to capture the emergent meso- (regional population ensemble) and macro-scale (whole brain) behavior, which is largely deterministic and reproducible across individuals. Indeed, recent modeling effort based on spectral graph theory has shown that an analytical model without regionally varying parameters can capture the empirical magnetoencephalography frequency spectra and the spatial patterns of the alpha and beta frequency bands accurately. In this work, we demonstrate an improved hierarchical, linearized, and analytic spectral graph theory-based model that can capture the frequency spectra obtained from magnetoencephalography recordings of resting healthy subjects. We reformulated the spectral graph theory model in line with classical neural mass models, therefore providing more biologically interpretable parameters, especially at the local scale. We demonstrated that this model performs better than the original model when comparing the spectral correlation of modeled frequency spectra and that obtained from the magnetoencephalography recordings. This model also performs equally well in predicting the spatial patterns of the empirical alpha and beta frequency bands.

Laplacian eigenmaps manifold learning and anomaly detection methods for spectral images

Abstract: Laplacian Eigenmaps Manifold Learning and Anomaly Detection Methods for Spectral Images Marcela Munoz Reales Supervising Professor: Dr. William Basener Spectral images provide a large amount of spectral information about a scene, but sometimes when studying images, we are interested in specific components. It is a difficult problem to separate the relevant information or what we call interesting from the background of a spectral image, even more so if our target objects are unknown. Anomaly detection is a process by which algorithms are designed to separate the anomalous (different) points from the background of an image. The data is complex and lives in a high dimension, manifold learning algorithms are used to analyze data that lives in a high dimensional space, but that can be represented as a lower dimensional manifold embedded in the high dimensional space. Laplacian Eigenmaps is a manifold learning algorithm that applies spectral graph theory methods to perform a non-linear dimensionality reduction that preserves local neighborhood information. We present an

A Study on the Effect of Triads on the Wigner’s Semicircle Law of Networks

Abstract: Spectral graph theory is widely used to analyze network characteristics. In spectral graph theory, the network structure is represented with a matrix, and its eigenvalues and eigenvectors are used to clarify the characteristics of the network. However, it is difficult to accurately represent the structure of a social network with a matrix. We derived the Wigner’s semicircle law that appears in the universality for the eigenvalue distribution of the normalized Laplacian matrix representing the structure of social networks, and proposed the analysis method to apply the spectral graph theory to social networks using the Wigner’s semicircle law. In previous works, we assume that nodes in a network are connected independently. However, in actual social networks, there are dependent structures called triads where link connections cannot be independent. For example, a triad is generated when a person makes a new friend via the introduction by its friend. In this paper, we experimentally investigate the effect of triads on the Wigner’s semicircle law, and clarify how effectively the Wigner’s semicircle law can be used for the analysis of networks with triads.

A Further Study on the Degree-Corrected Spectral Clustering under Spectral Graph Theory

Abstract: Spectral clustering algorithms are often used to find clusters in the community detection problem. Recently, a degree-corrected spectral clustering algorithm was proposed. However, it is only used for partitioning graphs which are generated from stochastic blockmodels. This paper studies the degree-corrected spectral clustering algorithm based on the spectral graph theory and shows that it gives a good approximation of the optimal clustering for a wide class of graphs. Moreover, we also give theoretical support for finding an appropriate degree-correction. Several numerical experiments for community detection are conducted in this paper to evaluate our method.